Scinovex
Subscribe

Research article Β· 2018 Β· Full text

International Journal of Statistics and Applied Mathematics

On the property of properness in the extended triple-error-correcting BCH codes

Anita Pruthi

Volume 3(2) Β· Pages 45–47 Β· DOI: 10.22271/maths.2018.v3.i2a.602 β†—

Research areasCoding theory and cryptographyAdvanced Wireless Communication TechniquesError Correcting Code Techniques

↓ Read full-text PDF0 citations

Abstract

Author-written

The undetected error probability Pu (€) for the primitive triple-error-correcting BCH codes of blocklength 2m - 1 on a BSC with cross-over probability € ≀ 1/2 has been widely studied. In this correspondence, extended triple-error-correcting BCH codes of blocklength 2m are studied and the results presented in are supported and strengthened but with a different approach i.e., it is proved that the extended triple-error-correcting BCH codes of blocklength 2m don’t satisfy the properties of proper codes for even m β‰₯ 6, m integer.

Summary

Source-checked

Plain-language reading of the full paper, prepared with AI and checked against the text. Each point links to the authors' own words.

  1. Why it matters

    • Proper codes are extensively studied because they provide good performance in error controlling. View evidence β†—

      Why it matters

      Proper codes are extensively studied because they provide good performance in error controlling.

      In the paper's own words

      β€œProper codes are studied extensively, specifically in [3]. Performance of proper codes is good in error controlling.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—
    • A code is defined as 'proper' if its undetected error probability, denoted as 𝑝𝑒(πœ€), increases as the cross-over probability πœ€ increases. View evidence β†—

      Why it matters

      A code is defined as 'proper' if its undetected error probability, denoted as 𝑝𝑒(πœ€), increases as the cross-over probability πœ€ increases.

      In the paper's own words

      β€œA code is said to be proper if its undetected error probability 𝑝𝑒(πœ€) is an increasing function of πœ€.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—
    • Undetected errors occur when transmission errors result in a received vector that is itself a valid codeword, making the error impossible to detect. View evidence β†—

      Why it matters

      Undetected errors occur when transmission errors result in a received vector that is itself a valid codeword, making the error impossible to detect.

      In the paper's own words

      β€œBut it may also happen than errors are so combined that the received vector is also one of the codewords. In such a case there is no way to detect that received codeword is not the sent codeword and in this way there arises an undetected error.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—

  2. The question

    • The study aims to prove that extended triple-error-correcting BCH codes with blocklength 2π‘š are not proper codes for even integers π‘š β‰₯ 6. View evidence β†—

      The question

      The study aims to prove that extended triple-error-correcting BCH codes with blocklength 2π‘š are not proper codes for even integers π‘š β‰₯ 6.

      In the paper's own words

      β€œextended triple-error-correcting BCH codes of blocklength 2π‘š are studied and the results presented in are supported and strengthened but with a different approach i.e., it is proved that the extended triple-error-correcting BCH codes of blocklength 2π‘š don’t satisfy the properties of proper codes for even π‘šβ‰₯6, π‘š integer.”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—

  3. What they did

    • The undetected error probability 𝑝𝑒(πœ€) was calculated using the MacWilliams identity based on the weight distribution of the code's dual. View evidence β†—

      What they did

      The undetected error probability 𝑝𝑒(πœ€) was calculated using the MacWilliams identity based on the weight distribution of the code's dual.

      In the paper's own words

      β€œNow 𝑝𝑒(πœ€) for an (n, k) code can be calculated from the weight distribution of its dual by using the MacWilliams identity”

      Page 1

      Found in: Introduction

      Read it in the full textOpen PDF at page 1 β†—
    • The analysis evaluated the undetected error probability for values of π‘š ranging from 6 to 16. View evidence β†—

      What they did

      The analysis evaluated the undetected error probability for values of π‘š ranging from 6 to 16.

      In the paper's own words

      β€œTable 1: Values of undetected error probability 𝑝𝑒(πœ€) for given πœ€, for 6 β‰€π‘šβ‰€16.”

      Page 2

      Found in: Introduction

      Read it in the full textOpen PDF at page 2 β†—
    • The researchers used the derivative of the undetected error probability with respect to πœ€ (𝑑𝑝𝑒(πœ€)/π‘‘πœ€) to determine if the function was monotonically increasing. View evidence β†—

      What they did

      The researchers used the derivative of the undetected error probability with respect to πœ€ (𝑑𝑝𝑒(πœ€)/π‘‘πœ€) to determine if the function was monotonically increasing.

      In the paper's own words

      β€œOn differentiating 𝑝𝑒(πœ€) w.r.t.,”

      Page 2

      β€œAlso, on calculating 𝑑𝑝𝑒(πœ€)/π‘‘πœ€ is not always found positive for given range of πœ€ and different values of m as shown in Table 1.”

      Page 3

      Found in: Introduction

      Read it in the full textOpen PDF at page 2 β†—

  4. What they found

    • For π‘š=6 and πœ€=0.1, the derivative 𝑑𝑝𝑒(πœ€)/π‘‘πœ€ was negative (-8.250700739498e-02), indicating the probability was not increasing. View evidence β†—

      What they found

      For π‘š=6 and πœ€=0.1, the derivative 𝑑𝑝𝑒(πœ€)/π‘‘πœ€ was negative (-8.250700739498e-02), indicating the probability was not increasing.

      In the paper's own words

      β€œ1. | 6 | 4 | 0.1 | 9.837993990569e-07 | -8.250700739498e-02”

      Page 2

      Found in: Table 1

      Read it in the full textOpen PDF at page 2 β†—
    • The undetected error probability becomes constant for values of π‘š β‰₯ 10. View evidence β†—

      What they found

      The undetected error probability becomes constant for values of π‘š β‰₯ 10.

      In the paper's own words

      β€œMoreover, for π‘šβ‰₯10, it becomes constant.”

      Page 3

      Found in: Conclusion

      Read it in the full textOpen PDF at page 3 β†—
    • The bound 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) < 2βˆ’π‘ is only satisfied for π‘š=6; for π‘š= 8, 10, 12, and 14, the bound is exactly equal to 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯). View evidence β†—

      What they found

      The bound 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) < 2βˆ’π‘ is only satisfied for π‘š=6; for π‘š= 8, 10, 12, and 14, the bound is exactly equal to 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯).

      In the paper's own words

      β€œTable 2 shows that 𝑝𝑒(πœ€) doesn’t satisfies 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) < 2βˆ’π‘ bound except for π‘š= 6. For π‘š= 8, 10, 12, 14, … .. the bound is exactly equal to 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯).”

      Page 3

      Found in: Conclusion

      Read it in the full textOpen PDF at page 3 β†—

  5. What it means

    • The authors conclude that extended primitive triple-error-correcting BCH codes of blocklength 2π‘š are not proper for even integers π‘š β‰₯ 6. View evidence β†—

      What it means

      The authors conclude that extended primitive triple-error-correcting BCH codes of blocklength 2π‘š are not proper for even integers π‘š β‰₯ 6.

      In the paper's own words

      β€œDue to the above mentioned observations, the code doesn’t exhibit the property of properness. This completes the proof that the extended primitive triple-error-correcting BCH codes of Blocklength 2π‘š are not proper for even π‘šβ‰₯6, π‘š integer.”

      Page 3

      Found in: Conclusion

      Read it in the full textOpen PDF at page 3 β†—

Visual summary

1 visuals

Illustrations explain the paper; they do not show study data.

AI-generated illustration
AI-generated conceptual illustration (Abstract mathematical functions represented as geometric curves on a coordinate plane). It explains the idea and does not show study data.β†— Enlarge
Concept of Properness
AI-generated conceptual illustration (Abstract mathematical functions represented as geometric curves on a coordinate plane). It explains the idea and does not show study data.

AI-generated conceptual illustration (Abstract mathematical functions represented as geometric curves on a coordinate plane). It explains the idea and does not show study data.

Open full-size image β†—

Concept of Properness

AI-generated conceptual illustration (Abstract mathematical functions represented as geometric curves on a coordinate plane). It explains the idea and does not show study data.

View source evidence β†—

Figures & tables

From the original paper
Values of undetected error probability 𝑝𝑒(πœ€) for given πœ€, for 6 β‰€π‘šβ‰€16.Table 1Page 2Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘Table 2Page 3Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘Table 2Page 3Undetected error probability for π‘š= 6, 8 and 10.Figure 1Page 3
FULL TEXT Original article Β· reflowed for reading

Introduction

Proper codes are studied extensively, specifically in [3]. Performance of proper codes is good in error controlling. A code is said to be proper if its undetected error probability 𝑝𝑒(πœ€) is an increasing function of πœ€. Proper codes perform good in error control [4].

When a codeword from a code is transmitted over a channel and some error occurs during transmission and if the received vector is not a codeword then errors are detected. But it may also happen than errors are so combined that the received vector is also one of the codewords. In such a case there is no way to detect that received codeword is not the sent codeword and in this way there arises an undetected error. 2. Claim: The extended triple-error-correcting BCH codes of blocklength 2π‘š are not proper for even π‘šβ‰₯6, π‘š integer. 3.

Proof: For a code to be proper its undetected error probability 𝑝𝑒(πœ€) should increase as πœ€ (Bit error rate) increases. Now 𝑝𝑒(πœ€) for an (n, k) code can be calculated from the weight distribution of its dual by using the MacWilliams identity and its expression is given by 𝑝𝑒(πœ€) = 2βˆ’(π‘›βˆ’π‘˜) 𝐡(1 βˆ’2𝑝) βˆ’(1 βˆ’π‘)𝑛, where (1 βˆ’2𝑝) = βˆ‘ 𝐡𝑖(1 βˆ’2𝑝)𝑖 𝑛 𝑖=0 , 𝐡𝑖, 0 ≀𝑖≀𝑛, represents weight distribution of dual code, p is the transition probability of the BSC. Although, 𝑝𝑒(πœ€) can be calculated not only by the wright distribution of the dual of the code but also by the weight distribution of the code itself. So in this claim, 𝑝𝑒(πœ€) is calculated by using weight distribution of the dual of extended triple-error-correcting BCH codes of Blocklength 2π‘š.

The weight distribution [5] of the dual of the code is as follows: Weight i Number of Vectors Bi 0, 2π‘š 1 2π‘šβˆ’1 Β± 2(π‘š+2)/2 2π‘š(2π‘šβˆ’1)(2π‘šβˆ’4)/960 2π‘šβˆ’1 Β± 2π‘š/2 7. 2π‘š(2π‘šβˆ’1)/48 2π‘šβˆ’1 Β± 2(π‘šβˆ’2)/2 2. 2π‘š(2π‘šβˆ’1)(3. 2π‘š+ 8)/15 Anita Pruthi PG Department of Mathematics, D.A.V. College, Abohar, Punjab, India http://www.mathsjournal.com 2π‘šβˆ’1 (2π‘šβˆ’1)(29. 22π‘šβˆ’4. 2π‘š+ 64)/32 Putting 𝑙= 2(π‘šβˆ’2)/2 and 𝑝= 3π‘š+ 1 we have (i) On differentiating 𝑝𝑒(πœ€) w.r.t., (iii) Using Equations (i) and (ii), we have the following table:- Table 1: Values of undetected error probability 𝑝𝑒(πœ€) for given πœ€, for 6 β‰€π‘šβ‰€16. S. No. π’Ž 𝒍 𝜺 𝒑𝒖(𝜺) π’…π’…πœΊ ⁄ 𝒑𝒖(𝜺) 1. 6 4 0.1 9.837993990569e-07 -8.250700739498e-02 0.2 1.918934366274e-06 -4.945716694417e-05 0.3 1.907454876543e-06 -1.898542817751e-08 0.4 1.907348626480e-06 -4.937333414117e-13 0.5 1.907348632812e-06 1.084202172486e-19 2. 8 8 0.1 2.980232365649e-08 -5.470128317751e-10 0.2 2.980232238770e-08 -8.624408683311e-23 0.3 2.980232238770e-08 -1.222599721827e-39 0.4 2.980232238770e-08 -8.834101051736e-184 0.5 2.980232238770e-08 0.000000000000e+00 3. 10 16 0.1 4.656612873077e-10 -2.212545631085e-44 0.2 4.656612873077e-10 -1.066902703526e-233 0.3 4.656612873077e-10 0.0 0.4 4.656612873077e-10 0.0 0.5 4.656612873077e-10 0.0 http://www.mathsjournal.com 4. 12 32 0.1 7.275957614183e-12 0.0 0.2 7.275957614183e-12 0.0 0.3 7.275957614183e-12 0.0 0.4 0.5 7.275957614183e-12 7.275957614183e-12 0.0 0.0 5. 14 64 0.1 1.136868377216e-13 0.0 0.2 1.136868377216e-13 0.0 0.3 1.136868377216e-13 0.0 0.4 1.136868377216e-13 0.0 0.5 1.136868377216e-13 0.0 6. 16 128 0.1 1.776356839400e-15 0.0 0.2 1.776356839400e-15 0.0 0.3 1.776356839400e-15 0.0 0.4 1.776356839400e-15 0.0 0.5 1.776356839400e-15 0.0 Table 2: Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘ m 𝒑𝒖(πœΊπ’Žπ’‚π’™) πŸβˆ’π’‘ 6 1.907348632812446e-06 1.907348632812500e-06 8 2.980232238769531e-08 2.980232238769531e-08 10 4.656612873077393e-10 4.656612873077393e-10 12 7.275957614183426e-12 7.275957614183426e-12 14 1.136868377216160e-13 1.136868377216160e-13 16 1.776356839400250e-15 1.776356839400250e-15 linear codes, IEEE Trans.

Table 2. Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘
m𝒑 (𝜺 ) 𝒖 π’Žπ’‚π’™πŸβˆ’π’‘
61.907348632812446e-061.907348632812500e-06
82.980232238769531e-082.980232238769531e-08
104.656612873077393e-104.656612873077393e-10
127.275957614183426e-127.275957614183426e-12
141.136868377216160e-131.136868377216160e-13
161.776356839400250e-151.776356839400250e-15

Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.

View original table image
Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘β†— Enlarge
Table 2
Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘

Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘

Open full-size image β†—
Original paper tableSource PDF Β· page 3 β†—
Table 2. Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘
4.12320.17.275957614183e-120.0
0.27.275957614183e-120.0
0.37.275957614183e-120.0
0.4 0.57.275957614183e-12 7.275957614183e-120.0 0.0
5.14640.11.136868377216e-130.0
0.21.136868377216e-130.0
0.31.136868377216e-130.0
0.41.136868377216e-130.0
0.51.136868377216e-130.0
6.161280.11.776356839400e-150.0
0.21.776356839400e-150.0
0.31.776356839400e-150.0
0.41.776356839400e-150.0
0.51.776356839400e-150.0

Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.

View original table image
Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘β†— Enlarge
Table 2
Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘

Comparison of 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) and 2βˆ’π‘

Open full-size image β†—
Original paper tableSource PDF Β· page 3 β†—
Table 1. Values of undetected error probability 𝑝𝑒(πœ€) for given πœ€, for 6 β‰€π‘šβ‰€16.
S. No.π’Žπ’πœΊπ’‘ (𝜺) 𝒖𝒅 ⁄ 𝒑 (𝜺) π’…πœΊ 𝒖
1.640.19.837993990569e-07-8.250700739498e-02
0.21.918934366274e-06-4.945716694417e-05
0.31.907454876543e-06-1.898542817751e-08
0.41.907348626480e-06-4.937333414117e-13
0.51.907348632812e-061.084202172486e-19
2.880.12.980232365649e-08-5.470128317751e-10
0.22.980232238770e-08-8.624408683311e-23
0.32.980232238770e-08-1.222599721827e-39
0.42.980232238770e-08-8.834101051736e-184
0.52.980232238770e-080.000000000000e+00
3.10160.14.656612873077e-10-2.212545631085e-44
0.24.656612873077e-10-1.066902703526e-233
0.34.656612873077e-100.0
0.44.656612873077e-100.0
0.54.656612873077e-100.0

Table text extracted from the original PDF; superscript letters are shown inline. The original table image is below.

View original table image
Values of undetected error probability 𝑝𝑒(πœ€) for given πœ€, for 6 β‰€π‘šβ‰€16.β†— Enlarge
Table 1
Values of undetected error probability 𝑝𝑒(πœ€) for given πœ€, for 6 β‰€π‘šβ‰€16.

Values of undetected error probability 𝑝𝑒(πœ€) for given πœ€, for 6 β‰€π‘šβ‰€16.

Open full-size image β†—
Original paper tableSource PDF Β· page 2 β†—

Inform. Theory 1979;25(1):110-112. 4. Dodunekova R, Dodunekov SM, Nikolova E, A survey on proper codes, Discrete Applied Mathematics 2008;156(9):1499-1509. 5. Mac Williams, Sloane NJA.

Theory of error correcting codes, North-Holland Publishing Company, New York 1977. Fig 1: Undetected error probability for π‘š= 6, 8 and 10.

Figure 1. Undetected error probability for π‘š= 6, 8 and 10.
Undetected error probability for π‘š= 6, 8 and 10.β†— Enlarge
Figure 1
Undetected error probability for π‘š= 6, 8 and 10.

Undetected error probability for π‘š= 6, 8 and 10.

Open full-size image β†—
Original paper figureSource PDF Β· page 3 β†—

Conclusion

Table 2 shows that 𝑝𝑒(πœ€) doesn’t satisfies 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯) < 2βˆ’π‘ bound except for π‘š= 6. For π‘š= 8, 10, 12, 14, … .. the bound is exactly equal to 𝑝𝑒(πœ€π‘šπ‘Žπ‘₯). Also Figure 1 and Table 1 represents that the undetected error probability is not always monotonically increasing as required for proper code. Moreover, for π‘šβ‰₯10, it becomes constant.

Also, on calculating 𝑑𝑝𝑒(πœ€)/π‘‘πœ€ is not always found positive for given range of πœ€ and different values of m as shown in Table 1. Forπ‘šβ‰₯12, as 𝑝𝑒(πœ€) attains constant value, 𝑑𝑝𝑒(πœ€)/ π‘‘πœ€ vanishes. Due to the above mentioned observations, the code doesn’t exhibit the property of properness. This completes the proof that the extended primitive triple-error- correcting BCH codes of Blocklength 2π‘š are not proper for even π‘šβ‰₯6, π‘š integer.

References

3 in the original paper
  1. Ong CT, Leung C. On the undetected error probability of triple-error-correcting BCH codes, IEEE Trans. Inform. Theory 1991;37(3):673-678.
  2. Patrick Perry, Necessary condition for good error detection, IEEE Trans. Inform. Theory 1991;37(2):375- 378.
  3. Leung-Yan-Cheong SK, Barnes ER, Friedman DU. On some properties of the undetected error probability of